That should be fine in terms of serving the web app -- writing to your Dropbox share is normally as fast as writing to anywhere else in your file storage.
However, if you're depending on it synchronising up to Dropbox and then down to your own machines quickly it might not be ideal, as there are quite a lot of machines involved -- ours, Dropbox's and yours -- and things can sometimes take a while to sync up. You'd also need to look out for synchronisation clashes -- for example, if you had something that updated a file on PythonAnywhere and also on your own machines, Dropbox might get confused.